Determination and validation of the elastic moduli of small and complex biological samples: bone and keratin in bird beaks

In recent years, there has been a surge in the development of finite-element (FE) models aimed at testing biological hypotheses. For example, recent modelling efforts suggested that the beak in Darwin''s finches probably evolved in response to fracture avoidance. However, knowledge of the material properties of the structures involved is crucial for any model. For many biological structures, these data are not available and may be difficult to obtain experimentally given the complex nature of biological structures. Beaks are interesting as they appear to be highly optimized in some cases. In order to understand the biomechanics of this small and complex structure, we have been developing FE models that take into account the bilayered structure of the beak consisting of bone and keratin. Here, we present the results of efforts related to the determination and validation of the elastic modulus of bone and keratin in bird beaks. The elastic moduli of fresh and dried samples were obtained using a novel double-indentation technique and through an inverse analysis. A bending experiment is used for the inverse analysis and the validation of the measurements. The out-of-plane displacements during loading are measured using digital speckle pattern interferometry.     

Dynamic crushing behavior of 3D closed-cell foams based on Voronoi random model

Dynamic crushing responses of three-dimensional cellular foams are investigated using the Voronoi tessellation technique and the finite element (FE) method. FE models are constructed for such closed-cell foam structures based on Voronoi diagrams. The plateau stress and the densification strain energy are determined using the FE models. The effects of the cell shape irregularity, impact loading, relative density and strain hardening on the deformation mode and the plateau stress are studied. The results indicate that both the plateau stress and the densification strain energy can be improved by increasing the degree of cell shape irregularity. It is also found that the plastic deformation bands appear firstly in the middle of the model based on tetrakaidecahedron at low impact velocities. However, the crushing bands are seen to be randomly distributed in the model based on Voronoi tessellation. At high impact velocities, the “I” shaped deformation mode is clearly observed in all foam structures. Finally, the capacity of foams absorbing energy can be improved by increasing appropriately the degree of cell shape irregularity.     

Experimental and computational characterization of designed and fabricated 50:50 PLGA porous scaffolds for human trabecular bone applications

The present study utilizes image-based computational methods and indirect solid freeform fabrication (SFF) technique to design and fabricate porous scaffolds, and then computationally estimates their elastic modulus and yield stress with experimental validation. 50:50 Poly (lactide-co-glycolide acid) (50:50 PLGA) porous scaffolds were designed using an image-based design technique, fabricated using indirect SFF technique, and characterized using micro-computed tomography (μ-CT) and mechanical testing. μ-CT data was further used to non-destructively predict the scaffold elastic moduli and yield stress using a voxel-based finite element (FE) method, a technique that could find application in eventual scaffold quality control. μ-CT data analysis confirmed that the fabricated scaffolds had controlled pore sizes, orthogonally interconnected pores and porosities which were identical to those of the designed files. Mechanical tests revealed that the compressive modulus and yield stresses were in the range of human trabecular bone. The results of FE analysis showed potential stress concentrations inside of the fabricated scaffold due to fabrication defects. Furthermore, the predicted moduli and yield stresses of the FE analysis showed strong correlations with those of the experiments. In the present study, we successfully fabricated scaffolds with designed architectures as well as predicted their mechanical properties in a nondestructive manner.     

Porosity-dependence of elastic properties and ultrasonic velocity in polycrystalline alumina — a model based on cylindrical pores

Effective elastic moduli and ultrasonic velocity of materials having aligned cylindrical pores have been derived using a series expansion in terms of the difference between the upper and lower bounds of elastic moduli obtained by the variational method. The theoretical results for polycrystalline alumina agree well with the experimental data, confirming the suggestion of previous researchers that a matrix containing parallel cylindrical pores orientated perpendicular to the applied stress, provide a better model than a spherical one in describing the porosity-dependence of elastic moduli in sintered specimens.     

A simple viscoelastic model for soft tissues the frequency range 6-20 MHz

In this paper, we present measurements of the shear properties of porcine skeletal muscle, liver, and kidney and a novel model describing them. Following a previously used method, shear mechanical impedances are measured, and complex shear moduli are obtained in the frequency range 6-20 MHz. As indicated in previous results, negative storage moduli are obtained in some measurements, which yield negative shear moduli in traditional linear viscoelastic models such as the Maxwell model, the Voigt model, and the Kelvin model. To resolve this problem, we propose a simple extension of the Voigt model. A mass is introduced into the model to account for the extra phase shift that apparently produces the negative moduli, and the shear stress thereby is related to the inertia of the material. The observed negative storage moduli are predicted by the new model when the relaxation time of the material is large and the working frequency is high. The model is fitted to experimental data to obtain values for material constants.     

A simple viscoelastic model for soft tissues in the frequency range 6-20 MHz.

In this paper, we present measurements of the shear properties of porcine skeletal muscle, liver, and kidney and a novel model describing them. Following a previously used method, shear mechanical impedances are measured, and complex shear moduli are obtained in the frequency range 6-20 MHz. As indicated in previous results, negative storage moduli are obtained in some measurements, which yield negative shear moduli in traditional linear viscoelastic models such as the Maxwell model, the Voigt model, and the Kelvin model. To resolve this problem, we propose a simple extension of the Voigt model. A mass is introduced into the model to account for the extra phase shift that apparently produces the negative moduli, and the shear stress thereby is related to the inertia of the material. The observed negative storage moduli are predicted by the new model when the relaxation time of the material is large and the working frequency is high. The model is fitted to experimental data to obtain values for material constants.     

Deformation in viscoelastic sandwich composites subject to moisture diffusion

This study analyzes the effect of moisture diffusion on the deformation of viscoelastic sandwich composites, which are composed of orthotropic fiber-reinforced laminated skins and viscoelastic polymeric foam core. It is assumed that the elastic and time-dependent (transient) moduli at any particular location in the foam core depend on the moisture concentration at that location. Sequentially coupled analyses of moisture diffusion and deformation are performed to predict overall performance of the studied viscoelastic sandwich systems. Time and moisture dependent constitutive model is used for the polymer foam core, while skins are assumed linear elastic. The overall time-dependent responses of the sandwich composites subject to moisture diffusion are analyzed using finite element (FE) method. Experimental data available in the literature and analytical solutions are used to support convergence studies in the FE analyses. Contributions of moisture dependent elastic and the time-dependent moduli to the overall stress, strain and displacement field are studied. FE analyses of the delamination between skins and core in sandwich composite under combined moisture diffusion and mechanical loading are also performed.     

Linear elasticity of planar delaunay networks. Part II: Voigt and Reuss bounds,and modification for centroids

Summary The linear elastic Delaunay network model developed in a previous paper is used to obtain further results on mechanical properties of graph-representable materials. First, we investigate the error involved in the uniform strain approximation — a computationally inexpensive approach widely employed in the determination of effective moduli of granular and fibrous media. Although this approximation gives an upper bound on the macroscopic moduli, it results in very good estimates of their second order statistics. In order to derive a lower bound another window definition has to be introduced. Also, an energy-based derivation of both bounds is given. The final result relates to a modification of a Delaunay network so that its vertices correspond to the centroids of cells of the corresponding Voronoi tessellation; an increase of effective moduli and a decrease of their scatter are observed.     

Hierarchical multiscale structure–property relationships of the red-bellied woodpecker (Melanerpes carolinus) beak

We experimentally studied beaks of the red-bellied woodpecker to elucidate the hierarchical multiscale structure–property relationships. At the macroscale, the beak comprises three structural layers: an outer rhamphotheca layer (keratin sheath), a middle foam layer and an inner bony layer. The area fraction of each layer changes along the length of the beak giving rise to a varying constitutive behaviour similar to functionally graded materials. At the microscale, the rhamphotheca comprises keratin scales that are placed in an overlapping pattern; the middle foam layer has a porous structure; and the bony layer has a big centre cavity. At the nanoscale, a wavy gap between the keratin scales similar to a suture line was evidenced in the rhamphotheca; the middle foam layer joins two dissimilar materials; and mineralized collagen fibres were revealed in the inner bony layer. The nano- and micro-indentation tests revealed that the hardness (associated with the strength, modulus and stiffness) of the rhamphotheca layer (approx. 470 MPa for nano and approx. 320 MPa for micro) was two to three times less than that of the bony layer (approx. 1200 MPa for nano and approx. 630 MPa for micro). When compared to other birds (chicken, finch and toucan), the woodpecker''s beak has more elongated keratin scales that can slide over each other thus admitting dissipation via shearing; has much less porosity in the bony layer thus strengthening the beak and focusing the stress wave; and has a wavy suture that admits local shearing at the nanoscale. The analysis of the woodpeckers'' beaks provides some understanding of biological structural materials'' mechanisms for energy absorption.     

Biomechanical evaluation of porous bioactive ceramics after implantation: micro CT-based three-dimensional finite element analysis

Hydroxyapatite ceramics have been widely investigated for bone regeneration due to their high biocompatibility. However, few studies focus on their mechanical characteristics after implantation. In this study, the finite element (FE) method was used to evaluate the mechanical properties of a fully interconnected porous hydroxyapatite (IPHA) over time of implantation. Based on the micro-CT images obtained from the experiments dealing with IPHA implanted into rabbit femoral condyles, three-dimensional FE models of IPHA (1, 5, 12, 24, and 48 weeks after implantation) were developed. FE analysis indicated that the elastic modulus gradually increased from 1 week and reached the peak value at 24 weeks, and then it kept at high level until 48 weeks postoperatively. In addition, as a local biomechanical response, strain energy density became to distribute evenly over time after the implantation. Results confirmed that the mechanical properties of IPHA are strongly correlated to bone ingrowth. The efficiency of the proposed numerical approach was validated in combination with experimental studies, and the feasibility of applying this approach to study such implanted porous bioceramics was proved.     

Closed-form effective elastic constants of frame-like periodic cellular solids by a symbolic object-oriented finite element program

In this study, the effective elastic constants of several 2D and 3D frame-like periodic cellular solids with different unit-cell topologies are analytically derived using the homogenization method based on equivalent strain energy. The analytical expressions of strain energy of a unit cell under different strain modes are determined using a generic symbolic object-oriented finite element (FE) program written in MATLAB. The obtained analytical expressions of the strain energy are then used to symbolically compute the effective elastic constants that include Young’s moduli, Poisson’s ratios, and shear moduli. The obtained analytical effective elastic constants are numerically verified using results from an ordinary numerical FE program. The obtained closed-form effective elastic constants are also compared with some existing solutions from the literature. This study demonstrates that symbolic computation platforms can be properly used to provide efficient methodologies for finding useful analytical solutions of mechanical problems. Without the symbolic object-oriented FE program in this study, elaborate and tedious analytical analysis has to be manually performed for each different unit cell. The symbolic object-oriented FE program provides analytical analysis of unit cells that is accurate and fast. The object-oriented programming technique allows the symbolic FE program in this study to be efficiently implemented.     

Modelling the deformation of a confectionery wafer as a non-uniform sandwich structure

The aim of this research was to model the mechanical behaviour of wafers found in various confectionery products in order to optimise the manufacturing stage. Compression and bending tests showed that the mechanical behaviour of the wafer was characteristic of a brittle foam. The internal microstructure of the wafer sheet was examined with an optical microscope which showed that the wafer possessed a sandwich structure with a porous core between two denser skins. An analytical model was developed to calculate the individual moduli of the wafer core and skin sections. These modulus values were used in a finite element (FE) model which consisted of a simple repetitive geometry. The FE model simulated the linear deformation of the wafer under compression and bending. The predictions from the analytical and numerical models were compared. They were found to agree in compression but deviated under bending due to the large mismatch of the core and skin moduli.     

Finite element modelling of core thickness effects in aluminium foam/composite sandwich structures under flexural loading

This paper models the flexural behaviour of a composite sandwich structure with an aluminium foam core using the finite element (FE) code LS-DYNA. Two core thicknesses, 5 and 20 mm, were investigated. The FE results were compared with results from previous experimental work that measured full-field strain directly from the sample during testing. The deformation and failure behaviour predicted by the FE model compared well with the behaviour observed experimentally. The strain predicted by the FE model also agreed reasonably well with the distribution and magnitude of strain obtained experimentally. However, the FE model predicted lower peak load, which is most likely due to a size effect exhibited by aluminium foam. A simple modification of the FE model input parameters for the foam core subsequently produced good agreement between the model and experimental results.     

偏心钢结构节点梁柱-核心区受力机理研究

某些工程处于建筑布置和使用的需要而采用钢结构偏心连接节点。偏心节点由于受到扭转作用,而在节点核心区产生较为明显的应力集中。该文借助有限元模型,对偏心连接梁-柱钢结构节点的受力特征进行分析,表明偏心节点核心区存在明显的三维扭转效应,以及梁柱截面的畸变翘曲效应。在受力特征分析的基础上,提出了钢结构偏心节点核心区的耦合效应计算模型。通过与有限元模型进行比对验证和参数分析,对梁柱截面畸变角函数进行修正和标定。分析表明,该文提出的偏心节点核心区耦合效应分析方法能够较准确计算梁柱截面畸变翘曲正应力与弯曲正应力,可供设计参考。     

Noise reduction capability of hybrid flax fabric-reinforced polypropylene-based composites

In recent years, natural fiber polymer composites have demonstrated great potential for reducing noise. This study reports sound transmission loss (STL) of hybrid flax fabric-reinforced polypropylene composites. The STL values were determined by the BSWA impedance tube and evaluated by ISO 10534 standard test procedure in the frequency range of 64 to 1600 Hz. Measurements show that the STL per areal density of flax/polypropylene is two times higher compared to aluminum. A noteworthy result is that upon hybridization with 13% carbon fibers (carbon-flax/polypropylene), the STL per areal density increases by 2.4 times. This study also presents the finite element (FE) simulation of sound transmission through the composites. Good agreement between FE predicted and experimentally measured STL is obtained. Moreover, parametric sensitivity studies are performed using the developed FE model to examine the effect of possible variation in the mechanical properties of the natural fiber composites on the STL performance. It is found that within 15% variation, no significant effect on STL is obtained and that STL is only sensitive to the in-plane elastic moduli.     

An approach to modeling time-dependent creep and residual stress relaxation around cold worked holes in aluminium alloys at room temperature

Creep behaviour of aluminium alloys is also observed at room temperature. As a result, a relaxation occurs of deliberately introduced beneficial residual stresses around fastener holes, before the relevant structural component is subjected to exploitation. Therefore, to adequately asses the life-time of the component with cold worked holes, it is necessary to quantify this relaxation. In this paper a combined iterative approach for building a time-dependent creep constitutive model of aluminium alloys at room temperature has been developed in order to be used in finite element (FE) simulations of the cold hole working process. The approach is based on an experimental study of the change in diameters of cold worked holes through mandrel cold working method and a subsequent series of FE simulations of the cold working process and of the following creep behaviour to determine the necessary equivalent stresses in the constitutive model. The obtained creep constitutive model has been founded on the power-law model. The model parameters A, n and m have been determined on the basis of a developed by the authors algorithm. The approach has been illustrated on D16T aluminium alloy widely used in the airspace industry. The material behaviour in the plastic field has been described by the nonlinear kinematic hardening model, obtained through a uniaxial tensile test. Both constitutive models have been used in FE simulations of the cold working processes and of subsequent residual stress relaxation around the cold worked open holes due to creep at room temperature. On the base of the FE results, mathematical models describing the residual stress relaxation have been obtained. Thus, the residual stresses are adequately evaluated immediately before introducing the structural component in operation.     

Approximation of curved cracks under mixed-mode loading

The calculation of stress intensity factors or mechanical energy release rate for non-straight cracks can be complicated. Approximation to equivalent crack shapes can simplify calculations considerably, but this requires an understanding of the influence of key shape parameters on crack-tip stresses. A simple analytical model has been developed, based on the concept of a relaxed volume, to predict mechanical energy release rate and deflection angle for a range of crack shapes under mixed-mode loading. Results from this model compared well with those obtained from finite element (FE) simulations, and with predictions from previous analytical models. It was found that the crack length and orientation of the crack-tip with respect to loading direction are the key influences on fracture parameters, whilst curvature near the crack-tip can also affect results.     

Modeling loading rate effect on crushing stress of metallic cellular materials

A mesoscale numerical model based on Voronoi tessellation is developed to investigate the loading rate effect on the crushing stress of cellular materials. The crushing stresses at both the impact and stationary ends of the Voronoi structures are simulated. The influences of the impact velocity, specimen size, inertia, and rate dependence of the base material on the crushing stress are discussed. The underlining reason for the argument on the rate dependence of cellular materials is clarified by comparing the current simulation results with the numerical results based on a continuum model and a shock wave theory. The conflicting observations from previous experimental studies on the dynamic behavior of cellular materials by different researchers are explained by the simulation results as well as the shock wave theory.     

The toucan beak: Structure and mechanical response

The structure and mechanical response of a Toco toucan (Ramphastos toco) beak were established. The beak was found to be a sandwich composite with an exterior of keratin scales (50 μm diameter and 1 μm thickness) and a core composed of fibrous network of closed-cells made of collagen. The tensile strength of the external shell is about 50 MPa. Micro- and nanoindentation hardness measurements corroborate these values. The keratin shell exhibits a strain-rate sensitive response with a transition from slippage of the scales due to release of the organic glue, at a low strain rate (5 × 10^− 5 s^− 1) to fracture of the scales at a higher strain rate (1.5 × 10^− 3 s^− 1). The closed-cell foam consists of fibers having a Young's modulus (measured by nanoindentation) of 12.7 GPa. This is twice as high as the keratin shells, which have E = 6.7 GPa. This is attributed to their higher calcium content. The compressive collapse of the foam was modeled by the Gibson–Ashby constitutive equations.There is a synergistic effect between foam and shell evidenced by a finite-element analysis. The foam stabilizes the deformation of the keratin shell by providing an internal support which increases its buckling load under compressive loading.